2023 SSMO Speed Round Problems/Problem 2
Problem
Let ,
,
be independently chosen vertices lying in the square with coordinates
,
,
, and
. The probability that the centroid of triangle
lies in the first quadrant is
for relatively prime positive integers
and
Find
Solution
Let have coordinates
,
have coordinates
,
and
have coordinates
.
Note that all these coordinates are uniformly distributed between
and
.
Thus, we want to find the probability that
and
both hold, which are independent events.
If , then
.
Thus, there exists a bijection between when
and when
.
(The case of
occurs with probability
).
so the probability is
for the chance
.
Thus, the answer is thus